41.

Let f(x) = ${\int }_{\mathrm{u}\left(\mathrm{x}\right)}^{\mathrm{v}\left(\mathrm{x}\right)}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$. Then, f'(x) is equal to

• g[v(x)] - g[u(x)]

• g'[v(x)] - g'[u(x)]

• g[v(x)] v'(x) - g[u(x)] u'(x)

• None of the above

The value of ${\int }_0^{\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|\mathrm{dx}$ is

• 1

• 2

• 0

• 3

$\int \frac{\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)}{\sqrt{\left(1 + \sin \left(2\mathrm{x}\right)\right)}}\mathrm{dx}$ equals

• $\log \left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)$

• x

• $\log \left(\sin \left(\cos \left(\mathrm{x}\right)\right)\right)$

• $\log \left(\mathrm{x}\right)$

Area common to the circle x² + y² = 64 and the parabola y² = 12x is equal to

• $\left(\frac{16}{3}\right)\left(4\mathrm{\pi } + \sqrt{3}\right)$

• $\left(\frac{16}{3}\right)\left(8\mathrm{\pi } - \sqrt{3}\right)$

• $\left(\frac{16}{3}\right)\left(4\mathrm{\pi } - \sqrt{3}\right)$

• None of these

Area bounded by the curve y = x³, the x-axis and the ordinates x = - 2 and x = 1 is

• - 9

• $- \frac{15}{4}$

• $\frac{15}{4}$

• $\frac{17}{4}$

Differential equation of the family of curve y = a cos(µx) + b sin(µx), where a, b are arbitrary constants, is given by

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{\mu y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\mathrm{\mu }}^2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - {\mathrm{\mu }}^2\mathrm{y} = 0$

• None of these

The differential equation of all circles which pass through the origin and whose centres lie on y-axis is

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = 0$

The differential equation of the family of curve y = Ae^3x + Be^5x, where A, B are arbitrary constants, is

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 8\frac{\mathrm{dy}}{\mathrm{dx}} + 15 = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 8\frac{\mathrm{dy}}{\mathrm{dx}} + 15 = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

• None of these

Solution of the differential equation $\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{y} + \sqrt{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)}$ is

• $\mathrm{y} - \sqrt{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)} = {\mathrm{cx}}^2$

• $\mathrm{y} + \sqrt{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)} = {\mathrm{cx}}^2$

• $\mathrm{x} + \sqrt{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)} = {\mathrm{cy}}^2$

• $\mathrm{x} - \sqrt{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)} = {\mathrm{cy}}^2$

Using trapezoidal rule and taking n = 4, the value of ${\int }_0^2\frac{\mathrm{dx}}{1 + \mathrm{x}}$ will be

• 1.1167

• 1.1176

• 1.118

• None of these